EPJ Web of Conferences 92, 021 0 0 (2015)
DOI: 10.1051/epjconf/ 201 5 9 2 021 0 0
C Owned by the authors, published by EDP Sciences, 2015
Modelling fluid flow in a reciprocating compressor
Jan Tuhovcak1,a Jií Hejík1 and Miroslav Jícha1
1
Brno University of Technology, Faculty of Mechanical Engineering, Energy Institute, Technicka 2896/2 Brno 616 69,
Czech Republic
Abstract. Efficiency of reciprocating compressor is strongly dependent on the valves characteristics, which
affects the flow through the suction and discharge line. Understanding the phenomenon inside the compressor
is necessary step in development process. Commercial CFD tools offer wide capabilities to simulate the flow
inside the reciprocating compressor, however they are too complicated in terms of computational time and
mesh creation. Several parameters describing compressor could be therefore examined without the CFD
analysis, such is valve characteristic, flow through the cycle and heat transfer. The aim of this paper is to show
a numerical tool for reciprocating compressor based on the energy balance through the cycle, which provides
valve characteristics, flow through the cycle and heat losses from the cylinder. Spring-damping-mass model
was used for the valve description. Boundary conditions were extracted from the performance test of 4cylinder semihermetic compressor and numerical tool validation was performed with indicated p-V diagram
comparison.
1 Introduction
Reciprocating compressors are commonly used in many
technical fields, from household appliances to large-scale
compressors used in refineries and gas engineering.
Reducing compressor energy consumption by improving
the efficiency is a relevant step in decreasing the global
energy consumption. To achieve this goal it is necessary
to understand processes inside the compressor: heat flux
and fluid flow. Analysis according to [1] shows that
contribution to overall electric energy consumption in US
reaches up to 8 %. Statistics from industry sector are not
known, however expectation are even higher. Overall
efficiency of compressor consist of three main subefficiencies: electrical, mechanical and thermodynamic
efficiency. Thermodynamic efficiency is the lowest (80 –
83 %), therefore there is significant effort to improve
parts of compressor, which have the biggest influence on
the efficiency: pressure losses in suction and discharge
line, superheating of gas and cylinder leakage. Three
dimensional (3D) numerical tools (CFD) offer the most
detailed method for thermodynamic analysis of
compressor, however they still have high demands on
computational power and time, even despite progress in
computer science in last years. As this approach is not
suitable in development process, several simplified
models were developed to analyze reciprocating
compressor. Basically, there are three approaches used in
analysis of compressors. Quasi-static (0D) simulation
tool uses the energy balance over a control volume [2].
More complicated numerical tool was introduced by [3],
a
solving Euler equations over a cross section
perpendicular to piston in 1D model or over the height of
the cylinder in 2D model. The last option of zooming into
a compressor is abovementioned numerical tool based on
solving Navier-Stokes equation over a control volume. In
this paper 0D model is presented for in-cylinder analysis
of the compressor.
2 The model
Analysis of reciprocating compressor with 0D model,
presented in this paper, can be used to evaluate following
phenomena:
• Pressure history in a cylinder
• Valve dynamics and motion
• Pressure losses of a valve
The energy analysis of unsteady flow serve as a base
for 0D model. Flow inside the cylinder is neglected and
pressure is considered as uniform in the cylinder as well
as temperature, calculated using of the 1st Law of
Thermodynamics,
dW + dQ + ¦ dmi ⋅ hi = dU .
(1)
i
Q stands for heat transfer from/to the cylinder. Inflow
or outflow through the valves is represented by dmihi and
the change of inner energy is dU. The mass balance in the
control volume is described by eq. (2), where mcyl is mass
inside the cylinder and mi is mass transported over valves.
dmcyl
+ ¦ mi = 0 .
(2)
dt
i
Corresponding author: [email protected]
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Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20159202100
EPJ Web of Conferences
2.1 Crank mechanism
1
The work of the piston, dW, is done by the change of a
volume, which is determined by a piston motion. Using
the properties of the crank mechanism: length of the
crank lever r, the length of the piston rod l and the
smallest distance between a piston and a cylinder S0
(derived from the clearance volume), movement of the
piston could be solved for present angle of crankshaft in
time , eq. (3).
§ p ·κ
m = φeff ⋅ ρ1 ⋅ ¨ 2 ¸
© p1 ¹
κ −1
§
·
2κ p1 ¨ § p2 · κ ¸ .
1− ¨ ¸
κ − 1 ρ1 ¨¨ © p1 ¹ ¸¸
©
¹
(7)
Effective flow area eff includes besides actual flow
area, determined from valve position as a geometry
function, also local pressure loss coefficient . All other
parameters have their usual meaning.
2.4 Heat Transfer
2
§r·
S = r + l + S0 - r ⋅ cos α - l l - ¨ ¸ sin 2 α .
©l¹
BDC
r
'
TDC
&
l
(3)
S
Figure 1. Crank mechanism parameters.
2.2 Mathematical model of valves
Valves inside the compressor are operated by the forces
acting on it. A mathematical model was introduced by [4]
as a mass-spring system, examined more deeply in the
work of [5]. Movement of the valve can be described by
single degree of freedom equation
m ⋅ x + d ⋅ x + k ⋅ x = ¦ Fi ,
(4)
i
x( n −1) = −e ⋅ x( n ) .
(5)
Mass of the valve m and stiffness of the spring k are
basic properties of the spring system, however the
damping constant d must be determined experimentally,
as it is formed not just by the system, but also with the
surrounding environment. Usually it is left from eq. (4) as
its value is negligible [3]. When the valve hits the limiter,
it loses part of the kinetic energy, expressed by restitution
coefficient e in eq. (5). Significant contribution to the
force acting on valve have pressure difference over the
valve, stiffness of the spring and adhesive forces due to
oil between valve surface and seating. Adhesive force
Fadh causes delay in valve opening and its approximated
value could be obtained by using eq.(6) [3].
x
Fadh = f ⋅ 3 .
(6)
x
Parameter f stands for geometry properties of the
valve and seating. Equation (4) is solved by using 4th
order Runge-Kutta method.
Heat transfer in the cylinder influences the overall
efficiency of compressor significantly. Heating up the
working fluid at the beginning of compression process by
1 K will lead to reduction of COP by 0.32 % [6]. Several
approaches were used in relevant literature to evaluate
heat transfer coefficient correctly, however they seem to
under-predict or over-predict the value, especially during
the suction or discharge process. Most significant of them
were analyzed by Pereira [7], although none of them
agreed with his CFD simulation of simplified
axisymmetric domain. All approaches were based on
correlation of Nusselt number with different constants a,
b and c,
Nu = a Reb Pr c .
(8)
Determining the velocity inside the compressor plays
important role in the compressor heat transfer model. The
velocity is given by a piston speed during the
compression. Discharge and suction process are more
complicated as the velocity of the fluid behind the valve
is much higher than in piston speed and is strongly
affected by manifold geometry. Disconzi [8] suggested to
divide the compressor cycle in four processes: intake,
compression, expansion and exhaust.
Table 1. Reynolds number and constants for processes inside
the cylinder
Process
Re =
Compression
Discharge
Re =
2.3 Flow through the valve
Re =
ρ ( t ) DVP
μ(t )
ρ ( t ) DVP
+ VP 0.8VC (t ) 0.2
μ(t )
Re =
Expansion
Suction
Fliegner’s equation is used to calculate flow through the
nozzle. The flow is considered as stationary, one
dimensional and isentropic.
Reynolds number
ρ ( t ) DVP
ρ ( t ) DVP
μ(t )
+ VP −0.4VC ( t )1.4
μ(t )
Constants
a
b
c
0.08
0.8
0.6
0.08
0.8
0.6
0.12
0.8
0.6
0.08
0.9
0.6
For each process he proposed new correlation for heat
transfer based on numerical simulation (Table 1). New
correlation was also proposed to calculate velocity used
in Reynolds number. During the suction and discharge
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EFM 2014
process it is derived from the mass flow rate through the
valve:
m ( t )
.
(9)
Vc =
ρ( t ) ⋅ Ac
Where (t) is the density of gas in the cylinder and Ac
is the cross-sectional area of the cylinder. Vp stands for
the piston velocity. Table 1 shows new correlations from
the work of Disconzi. He noted that the highest amount
of heat is transferred during the suction due to the high
temperature difference between sucking gas and cylinder
walls. Also the heat transfer area is significantly larger
compared to discharge, which increases the importance of
correct heat transfer prediction in cylinder.
2.6 Compressor
The simulations were carried out for a real compressor,
from which we took geometry parameters and estimated
valve properties (Table 2). The working fluid for
compressors depends on its purpose. Ideal air was in
presented simulation tool to simplify the model in its
development phase. Boundary conditions (BC) were
obtained from real compressor measurements and also
geometry parameters are similar to the compressor.
Table 2. Compressor specification
2.5 Pressure waves in the cylinder
Distribution of pressure inside the cylinder is not
completely uniform and pressure waves occur when the
valves open and thus interact with the outflow valve.
Pressure waves are present in particular compressors type
as well as during re-expansion, even during the
compression process [3]. In reciprocating compressor
they cause negative effect on the crank mechanism,
exciting an oscillating moment onto the piston. Second
effect is associated with a valve movement. Analysis of
Aigner [3] concluded that valve dynamics is not correctly
described by conventional methods, when the pressure
waves are significant inside the cylinder. Impact velocity
of valves is overestimated, which is important fact in
design process of a compressor. Moreover the losses
through the valve are increased, reducing the overall
efficiency. However, when the piston velocity is small
compared to the velocity of sound in particular
environment and the motion is sufficiently smooth the
state of the gas in cylinder can be described by quasistatic change of state – 0D model [9]. Nevertheless, the
waves initiated by valves could be still present, especially
with compressing heavy gases in fast running
compressors. Therefore Steinrueck [9] introduced
dimensionless classification number * as a ratio of
characteristic time scales.
Δt
(10)
ε= w ,
Δt D
d
Δt w = ,
ci
(11)
S0
.
(12)
d S / dt 2
Time scale tw expresses the time that pressure wave
needs to pass through the diameter of the cylinder.
Characteristic dimension - diameter is chosen due to
valves located around the cylinder in this case. Using the
compressor with valves located on the top of the cylinder
changes diameter to the bore of the cylinder. Acceleration
of the piston in the top dead center (TDC), where it has
maximum, is represented by time scale tD. When the
classification number << 1, quasi-static model is valid.
Δt D =
2
Parameter
Dimension
Diameter of cylinder bore
Crank height
Rod length
Stroke
Crankshaft speed
Clearance volume
Valve weight
Spring stiffness
Damping constant
80 mm
28.75 mm
115 mm
57 mm
1450 rpm
3.78x103 mm3
0.05 kg
5000 N
0.1 Ns
When the pressure exerts a force on the valve, it
moves to certain position dependent on pressure
difference magnitude and the stiffness of the spring. Flow
area is calculated from the valve position and geometrical
function describing the shape of valve seat.
Boundary conditions (Table 3) were used as for real
compressor, extracted from the experiments.
Table 3. Boundary conditions used in the simulation tool
Boundary condition
Value
Suction pressure
Discharge pressure
Inlet temperature
4.36 bar
20.5 bar
20 °C
3 Methodology
The compressor simulation tool is solving the energy eq.
(1) transferred from differential form to difference
equation. Pressure inside the cylinder is linearly
approximated. Choosing the correct time step is essential
for simulation tool accuracy.
4 Results
The simulation tool offers an overview on compressor
working process and its efficiency. Different working
condition and compressor geometries can be evaluated
with the tool, however exact BC and valves description is
a necessary input. We used only estimated values as BC,
e.g. for valves, especially pressure loss coefficients for
presented results. Therefore the results are not
comparable with an actual compressor, however the
trends are correct.
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p / Pa
Fig. 2 shows the pressure behavior in the cylinder
over two crankshaft cycles. To overcome the stiffness of
the spring and the adhesive force of the oil the pressure
inside the cylinder raises up to 29 bars. Decrease in the
cylinder pressure during the suction process is not as
significant as for the discharge process due to the
different valve characteristic used in the model.
' / deg
Figure 2. Pressure history inside the cylinder.
valves is not present, which is not usual in real situations
and the reason can be found in the inaccurate valve
description. The pressure loss is changing with the valve
movement, however the model uses only one value for
pressure loss and the actual flow area is changing in time.
The heat transfer inside the cylinder was analyzed
using Diconzi’s correlation. He proposed a correlation for
each process inside the cylinder. Therefore the heat
transfer during the cycle has no smooth history. When the
process inside the cylinder changes, also the heat transfer
equation changes, causing sudden changes in the graph
(Fig. 3.) Overall behavior is similar to a real compressor,
however the solution shows inaccuracies. During the
suction the heat is transferred from the cylinder wall to
the gas, while during the latter phase of compression and
discharge the heat is transferred in opposite direction.
The temperature difference wall - gas is smaller during
the suction than during the discharge, however the heat
transfer surface is much bigger. The rapid change of
transferred heat during the suction process or discharge is
caused by mass flow coming to / from cylinder.
Temperature history is shown in the Fig. 4. As the gas
is compressed, an increase in the temperature is present.
After the re-expansion the temperature decreases even
under the suction temperature, what is theoretically
possible, but in reality not a usual state.
5 Conclusion
Q / W m-2
The 0D simulation tool for reciprocating compressor was
developed in the present study. The model is capable of
analyzing basic compressor parameters, evaluating the
valve behavior, pressure history and heat transfer in the
cylinder. Because of missing experimental data the model
was not validated for real compressor.
Acknowledgement
' / deg
This work was supported by the project
RP9042100306/1120 of Brno University of Technology –
Faculty of Mechanical Engineering and NETME Centre,
regional research and development center established and
founded from Operational Programme Research and
Development for Innovation in a frame of project
NETME Centre (New Technologies for Mechanical
Engineering), Reg. no.: CZ.1.05/2.1.00/01.0002 and
supported in a sustainability phase by project NETME
CENTRE PLUS (LO1202) financed by The Ministry of
Education, Youth and Sports in a frame of the “National
sustainability program I”.
T/K
Figure 3. Heat flux from/to the cylinder.
References
' / deg
Figure 4. Temperature history through the cycle.
Motion of valves could be evaluated also from the
Fig. 2. Valves are opened when the pressure before the
valve is higher than behind it. Valves are moving back to
the closed position rather fast and the oscillation of
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EFM 2014
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